Problem: Let $x$ be a positive real number such that $x + \frac{1}{x} = 98.$  Find\[\sqrt{x} + \frac{1}{\sqrt{x}}.\]
Answer: Let
\[y = \sqrt{x} + \frac{1}{\sqrt{x}}.\]Then
\[y^2 = x + 2 + \frac{1}{x} = 98 + 2 = 100.\]Since $\sqrt{x} \ge 0$ and $\frac{1}{\sqrt{x}} \ge 0,$ we must have $y \ge 0.$  Therefore, $y = \boxed{10}.$